Yes, Monaco still does the passport stamp. An email to the Monaco Tourism office said that I should go to the tourist information office in Monte Carlo (a couple of blocks from the casino) and that they were open on the weekends. If anyone is interested they should check the office's hours because they seemed to take a pretty long lunch and their hours might change based on the season.
Just like Philhayes said - upstairs at the train station at the tourist office you are able to get a passport stamp. I did not see any signs or mention of it anywhere in the tourist office but I asked the lady how to get to the tourist office by the casino and if they would stamp my passport. She gave me the instructions but also told me that she could provide the stamp right there. The easiest passport stamp I ever received.
Side note: sorry to hear about the Vatican not stamping passports. I was there a month ago and did not even think about it because I had read several years ago that they did not do passport stamps. I was unaware that they did passport stamps at one time. I imagine that it would have had some character to the stamp (unlike an EU rectangle stamp).
Thank you all for the assistance.
There are reports that at least one OECD Country -- Canada at least -- has sometimes considered "commemorative"passports stamps from say "tourism" offices to be a sign of damage to the passport and thus messing up passport renewal applications by persons who seek out and get the tourist stamp thing.